In particular, \(f_i\)  depends only on \(x_1\) ,..., \(x_i\) .
We now associate with a vector \(\lambda \in \mathbb {R}^n\)  the polytope \(P_I(\lambda )\)  defined by inequalities (cf. [1]}):
\(0\le x_1\le (\lambda ,\beta _1); \quad 0\le x_2\le -f_1(x_1)+(\lambda ,\beta _2);\quad 0\le x_3\le -f_2(x_1,x_2)+(\lambda ,\beta _3);\ \ldots \ ;\) 
\(0\le x_\ell \le -f_{\ell -1}(x_1,\ldots ,x_{\ell -1})+(\lambda ,\beta _{\ell }).\) 
